Kuzee
Infinite Series

Integral Test

Definition

Integral Test

Let f(x)f(x) be continuous, non-increasing, and non-negative on [1,+∞)[1, +\infty) with an=f(n)a_n = f(n). Then the series ∑n=1∞an\sum_{n=1}^{\infty} a_n and the integral ∫1+∞f(x)dx\int_1^{+\infty} f(x) dx share the same convergence behavior.

SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
∫\intMathematical symbolIntegral symbolUsed for definite or improper integrals

Formula

Integral Test

The series ∑n=1∞an\sum_{n=1}^{\infty} a_n and the integral ∫1+∞f(x)dx\int_1^{+\infty} f(x) dx converge or diverge together, where an=f(n)a_n = f(n) and f(x)f(x) is continuous, non-increasing, and non-negative on [1,+∞)[1, +\infty).

Applicable Scenarios

  • The general term can be represented by a continuous function.
  • Other convergence tests are difficult to apply.

Example

Example 1

Determine whether the series ∑n=1∞1nln⁡n\sum_{n=1}^{\infty} \frac{1}{n\ln n} converges.

Solution: Let f(x)=1xln⁡xf(x) = \frac{1}{x\ln x} so that an=f(n)a_n = f(n).

Evaluate the integral: ∫2+∞1xln⁡xdx=∫2+∞1ln⁡xd(ln⁡x)=ln⁡(ln⁡x)∣2+∞=+∞\int_2^{+\infty} \frac{1}{x\ln x} dx = \int_2^{+\infty} \frac{1}{\ln x} d(\ln x) = \ln(\ln x) \big|_2^{+\infty} = +\infty

The integral diverges, so the series diverges as well.

Exercises

Exercise 1

Determine whether the series ∑n=1∞1nln⁡n\sum_{n=1}^{\infty} \frac{1}{n\ln n} converges.

series convergenceintegral test

Approach: Apply the integral test to compare the series and the corresponding integral.

Steps:

  1. Let f(x)=1xln⁡xf(x) = \frac{1}{x\ln x} so that an=f(n)a_n = f(n).
  2. Evaluate the integral: ∫2+∞1xln⁡xdx=∫2+∞1ln⁡xd(ln⁡x)=ln⁡(ln⁡x)∣2+∞=+∞\int_2^{+\infty} \frac{1}{x\ln x} dx = \int_2^{+\infty} \frac{1}{\ln x} d(\ln x) = \ln(\ln x) \big|_2^{+\infty} = +\infty.
  3. The integral diverges, hence the series diverges.

Answer: The series diverges (divergence).

Summary

Symbols Used in This Article

SymbolTypePronunciation / NoteMeaning in This Article
f(x)f(x)Mathematical symbolFunctionA continuous function
ana_nMathematical symbolGeneral termThe nnth term of the series
ln⁡\lnMathematical symbolNatural logarithmThe natural logarithm function
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